College Algebra, 2013a

426 Exponential and Logarithmic Functions
y = f(x) =2log(3− x) − 1
( ) x
y = g(x) =ln
x − 1
While logarithms have some interesting applications of their own which you’ll explore in the exercises,
their primary use to us will be to undo exponential functions. (This is, after all, how they
were defined.) Our last example solidifies this and reviews all of the material in the section.
Example 6.1.5. Let f(x) =2 x−1 − 3.
1. Graph f using transformations and state the domain and range of f.
2. Explain why f is invertible and find a formula for f −1 (x).
3. Graph f −1 using transformations and state the domain and range of f −1 .
4. Verify ( f −1 ◦ f ) (x) =x for all x in the domain of f and ( f ◦ f −1) (x) =x for all x in the
domain of f −1 .
5. Graph f and f −1 on the same set of axes and check the symmetry about the line y = x.
Solution.
1. If we identify g(x) =2 x , we see f(x) =g(x − 1) − 3. We pick the points ( −1, 1 2)
,(0, 1)
and (1, 2) on the graph of g along with the horizontal asymptote y = 0 to track through
the transformations. By Theorem 1.7 we first add 1 to the x-coordinates of the points on
the graph of g (shifting g to the right 1 unit) to get ( 0, 1 2)
,(1, 1) and (2, 2). The horizontal
asymptote remains y = 0. Next, we subtract 3 from the y-coordinates, shifting the graph
down 3 units. We get the points ( 0, − 5 2)
,(1, −2) and (2, −1) with the horizontal asymptote
now at y = −3. Connecting the dots in the order and manner as they were on the graph of
g, we get the graph below. We see that the domain of f is the same as g, namely(−∞, ∞),
but that the range of f is (−3, ∞).
y
y
7
6
5
4
3
2
1
−3−2−1 −1 1 2 3 4
−2
−3
y = h(x) =2 x
x
−−−−−−−−−−−−→
7
6
5
4
3
2
1
−3−2−1 −1 1 2 3 4
−2
y = f(x) =2 x−1 − 3
x